Why? Because in addition to assuming a unique result, they are inconsistent. If:

\(1+2+3+ \cdots = -1/12 = A\)

Then for example consider this:

\(A-A = (1+2+3+ \cdots ) - (1+2+3+ \cdots )\)

which by rule 1:

\( 0 = (1+2+3+ \cdots ) - (0+1+2+3+ \cdots ) \)

such that

\(0 = 1+1+1+1+ \cdots\)

Aha! This now implies that

\(0=1 + (1+1+1+\cdots ) = 1+0\) and so \(0 = 1\)!!!!

The two rules work for alternating sums, but when the sign of the sum terms is the same the two rules are clearly inconsistent.

But does this mean that \(1+2+3+ \cdots \) is not \(-1/12\) ? Not at all. The result is still valid due to deeper reasons: analytic continuation of Riemann zeta function.

It is not easy to find why things like this work in math, but in general physics intuition is a very good clue that there must be a solid and rigorous foundation. It is just that physicists' focus is on solving the practical problems and not on the deeper mathematical theory. One may say that a physicist to a mathematician is like an engineer to a physicist :) This is not that bad though: the engineers make more money than physicists, and physicists make more money than mathematicians.

I still regard Mr. Bender's lectures as outstanding, but I should have trusted my mathematical intuition more and not disregarded the alarm bells of mathematical rigor. The inconsistent argument above is due to David Joyce and I ran across it on Quora where the -1/12 result is discussed often.